Born in Reus, Tarragona, Catalonia, Cuenca joined FC Barcelona's La Masia in 2003, aged just 12. Halfway through his third youth year, however, he returned home and joined local CF Reus Deportiu.

Cuenca finished his formation with CF Damm and Barcelona, rejoining the latter in 2009. He made his senior debuts in the 2010–11 season, being loaned to another Catalonia club, CE Sabadell FC in the third division; he made an immediate impact, starting in 24 of his appearances and totalling 2,288 minutes as the Arlequinats returned to the second division after an 18-year absence.

In June 2011 Cuenca returned to Barcelona and joined its B team, alongside Martí Riverola. Shortly after, he was called by first-team manager Pep Guardiola to the pre-season, and made his debut on 23 July against HNK Hajduk Split.

Cuenca made his official debut for Barça B on 4 September 2011, playing the last 29 minutes in a 4–0 away win against FC Cartagena, and scoring the last goal of the match. On 19 October he made his first official appearance for the main squad, replacing David Villa in the dying minutes of a 2–0 home success against FC Viktoria Plzeň for the season's UEFA Champions League.

On 25 October 2011, after another game as a substitute (unused), Cuenca made his La Liga debut, playing the full 90 minutes and being booked in a 1–0 away win against Granada CF. Four days later, again as a starter, he scored in a 5–0 home success over RCD Mallorca, assisted by Adriano.

On 3 December 2011 Cuenca netted his second league goal, playing the entire 5–0 home routing of Levante UD. On 31 January 2012 he signed a contract extension with the Blaugrana, keeping him at the club until June 2015, and also switched permanently to the first team, being assigned jersey number 23.